218 research outputs found

    Virtual Knot Theory --Unsolved Problems

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    This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.Comment: 33 pages, 7 figures, LaTeX documen

    Infinitely many two-variable generalisations of the Alexander-Conway polynomial

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    We show that the Alexander-Conway polynomial Delta is obtainable via a particular one-variable reduction of each two-variable Links-Gould invariant LG^{m,1}, where m is a positive integer. Thus there exist infinitely many two-variable generalisations of Delta. This result is not obvious since in the reduction, the representation of the braid group generator used to define LG^{m,1} does not satisfy a second-order characteristic identity unless m=1. To demonstrate that the one-variable reduction of LG^{m,1} satisfies the defining skein relation of Delta, we evaluate the kernel of a quantum trace.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-18.abs.htm

    On the Links-Gould invariant and the square of the Alexander polynomial

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    This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be seen as exterior powers of copies of Burau representations.Comment: 19 page

    A cubic defining algebra for the Links-Gould polynomial

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    We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure

    On a Relation Between ADO and Links-Gould Invariants

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    In this thesis we consider two knot invariants: Akutsu-Deguchi-Ohtsuki(ADO) invariant and Links-Gould invariant. They both are based on Reshetikhin-Turaev construction and as such share a lot of similarities. Moreover, they are both related to the Alexander polynomial and may be considered generalizations of it. By experimentation we found that for many knots, the third order ADO invariant is a specialization of the Links-Gould invariant. The main result of the thesis is a proof of this relation for a large class of knots, specifically closures of braids with five strands

    On the Colored HOMFLY-PT, Multivariable and Kashaev Link Invariants

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    We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.Comment: 16 page
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